Question

a. Obtain a point estimate for the mean tax efficiency of all mutual fund portfolios with \(6%\) of their investments in energy securities.

b. Determine a \(95%\) confidence interval for the mean tax efficiency of all mutual fund portfolios with \(6%\) of their investments in energy securities.

c. Find the predicted tax efficiency of a mutual fund portfolio with \(6%\) of its investments in energy securities.

d. Determine a \(95%\) prediction interval for the tax efficiency of a mutual fund portfolio with \(6%\)of its investments in energy securities.

Short answer

Part a. The conditional mean of the response variable corresponding to the predictor variable \(x=3\) is \(2\)

Part b. The \(95%\) confidence interval for the conditional mean is \(-0.013\) to \(4.013\)

Part c. The predicted value of the response variable corresponding to the predictor variable \(x=3\) is \(2\).

Part d. The \(95%\) prediction interval for the conditional mean is \(-2.930\) to \(6.930\).

Step-by-step solution

Part a. Step 1. Given information

Given,

\(x=3\)

Part a. Step 2. Calculation

Find the point estimate for the conditional mean of the response variable corresponding to the predictor variable, \(x=2\) by using MINITAB. MINITAB procedure:

Step 1: Choose Stat > Regression > Regression.

Step 2: In Response, enter the column y.

Step 3: In Predictors, enter the columns X.

Step 4: In Options, enter \(3\) under Prediction interval for new observations

Step 5: In Confidence Level, enter \(95\).

Step 6: In Storage, Choose Fits, Confidence limits, SEs of fits, and Prediction limits.

Step 7: Click OK

MINITAB output: Predicted Values for New Observations

From the MINITAB output, the point estimate for the conditional mean of the response variable corresponding to the predictor variable \(x=3\) is \(2\).

Part b. Step 1. Calculation

From the MINITAB output in part (a), the confidence interval for the conditional mean of the response variable corresponding to the predictor variable \(x=3\) is \((-0.013, 4.013)\).

Hence, the confidence interval for the conditional mean is \((-0.013, 4.013)\).

Part c. Step 1. Calculation

From the MINITAB output in part (a), the predicted value of the response variable corresponding to the predictor variable \(x=3\) is \(2\)

Hence, the predicted value of the response variable corresponding to the predictor variable \(x=3\) is \(2\).

Part d. Step 1. Calculation

From the MINITAB output in part (a), the prediction interval for the conditional mean of the response variable corresponding to the predictor variable \(x=3\) is \((-2.930, 6.930)\)

Hence, \(95%\) prediction interval for the conditional mean is \((-2.930, 6.930)\)